Dates: 19 (Mon) 1 pm to 21 (Wed) August, 2024
Venue: Lecture Room 109, 1st Floor of the Faculty of Science Bldg. 1, Chiba University [campus map] [access]
Registration: https://docs.google.com/forms/d/1vIwuAw9Rm7KJPRmFaVN4cmQN6rgWivCGgGNRjxIVqRg/edit
If you need financial support, please send a request via the registration form above. Note that we give priority to students & young researchers and may not accept all the requests. Deadline: 8 August (Thu) at noon (because the administration office closes between 11 and 16 August).
Updates (8/21):
We will have a second talk on the last day by Azuna Nishida.
Speakers:
Tomohiro Asano (Kyoto U.)
Hansol Hong (Yonsei U.)
Doğancan Karabaş (Kavli IPMU)
Yuki Koto (Academia Sinica)
Hayato Morimura (Kavli IPMU)
Azuna Nishida (Chiba U.)
Schedule: PDF
Title and Abstract:
Tomohiro Asano : Sheaf-theoretic wrapping and heavy subsets in cotangent bundles
Tamarkin provided a method to prove non-displaceability of subsets of cotangent bundles with respect to Hamiltonian isotopies using the microlocal sheaf theory. On the other hand, Entov-Polterovich introduced the concepts of partial symplectic quasi-states and heaviness/superheaviness as more precise concepts leading to non-displaceability.
In this talk, I will give a sheaf-theoretic characterization of heaviness with respect to a partial symplectic quasi-state defined by Monzner-Vichery-Zapolsky. For the proof, we use sheaf-theoretic wrapping functors.
Hansol Hong : SYZ mirror symmetry for log Calabi-Yau surfaces
Any log Calabi-Yau surface X can be expressed as a nontoric blowup of a toric surface up to a modification of the boundary divisor. Using this, one can construct a special Lagrangian torus fibration on X with nodal fibers, which results in the cluster structure on its SYZ mirror. I will first describe its mirror potential in terms of combinatorial data on the associated scattering diagram, and examine how its critical loci are affected by blowups. In particular, we will see that the blowup creates exactly one nondegenerate `geometric’ critical point of the potential, which is mirror to semisimplicity of the quantum cohomology of X. The talk is based on joint works with Sam Bardwell-Evans, Mandy Cheung, Yu-Shen Lin and Hyunbin Kim. [slides of the second talk]
Doğancan Karabaş : 1. Wrapped Fukaya category of plumbings via gluing methods
Kontsevich's homological mirror symmetry conjecture proposes an equivalence between the derived invariants of a symplectic manifold (A-side) and its mirror variety (B-side). When the A-side is represented by a Weinstein manifold, its invariant can be modelled as its wrapped Fukaya category, which can further be described by the dg category of wrapped microlocal sheaves defined on its skeleton.
In my talk, I will utilise the framework developed by Ganatra, Pardon, and Shende, along with categorical tools from arXiv:2109.03411 and arXiv:2405.03258 (joint work with Sangjin Lee), to explain how to compute wrapped Fukaya categories concretely in a local-to-global manner. As an application, I will compute the wrapped Fukaya category of plumbing spaces, which are constructed by combining cotangent bundles according to a given quiver. Moreover, in the case of plumbings of cotangent bundles of (n>2)-dimensional spheres, I will show that their wrapped Fukaya categories are Morita equivalent to Ginzburg dg algebras. This is joint work with Sangjin Lee (arXiv:2405.10783).
2. A computational approach to the homotopy theory of dg categories
The homotopy theory of differential graded (dg) categories plays a significant role in various fields, including algebraic geometry, representation theory, higher categories, and symplectic topology. In particular, understanding dg categories is crucial for formulating and interpreting homological mirror symmetry.
In this talk, I will present our approach to the homotopy theory of dg categories by establishing a cofibration structure, which can be viewed as a half-model structure. This structure enables a combinatorial description of derived constructions and offers computational advantages. This is joint work with Sangjin Lee (arXiv:2109.03411 and arXiv:2405.03258). Some key applications of our approach include:
1) Combinatorial description of homotopy colimits of dg categories, which gives the formulas for computing wrapped Fukaya categories in the first talk,
2) Local-to-global construction of functors between wrapped Fukaya categories that are induced by symplectomorphisms,
3) A simple description of internal Hom and Hochschild cohomology of dg categories. This ongoing work aims to provide useful tools for addressing the Weinstein conjecture, which concerns the existence of periodic orbits of Reeb vector fields.
I plan to cover as much of this content as time permits, and according to the audience's interest.
Yuki Koto : Quantum cohomology of toric bundles
Quantum cohomology is a Frobenius manifold in the A-model. It can be studied by certain generating functions of genus zero Gromov-Witten invariants, which are called I-functions. For example, we can explicitly describe an I-function of a toric variety as a cohomology-valued hypergeometric series associated with its fan. Brown observed its relative version. He gave an explicit formula of an I-function of a (split) toric bundle constructed as a fiberwise GIT quotient of a direct sum of line bundles by a torus as a hypergeometric modification of an I-function of its base in terms of the fan of its fiber. In this talk, I will explain a generalization of Brown's result to non-split toric bundles. This work is partially based on joint work with Hiroshi Iritani.
Hayato Morimura : Sectorial covers over fanifolds
Sectorial descent developed by Ganatra--Pardon--Shende is a cosheaf property of partially wrapped Fukaya categories. It gives us a powerful computational tool, when stopped sectors admit a Weinstein sectorial cover. However, construction of such a cover is in general nontrivial. Also, in the context of HMS it should be compatible with the corresponding cover on the B-side so that local equivalences glue to yield global one. In this talk, we will explain background materials and ideas needed to construct a nice Weinstein sectorial cover of the stopped sector associated with any fanifold, a base over which Gammage--Shemde established HMS and which encodes combinatorial data to glue local mirror pairs.
Azuna Nishida : HMS for weighted projective spaces based on SYZ
Organizers:
Masahiro Futaki (Chiba U.)
Hiroshige Kajiura (Chiba U.)
Atsushi Kanazawa (Waseda U.)
Supported by:
JSPS Grant-in-Aid (C) 23K03084 (Hiroshige Kajiura)
JSPS Grant-in-Aid (C) 22K03282 (Masahiro Futaki)
JSPS Grant-in-Aid (C) 22K03296 (Atsushi Kanazawa)
If you have any questions, email me at (my family name) at math.s.chiba-u.ac.jp
Last modified on 22 August 2024 by Masahiro Futaki.